Transcript Slide 1

Nonlinear Systems:
Properties and Tests
M. Sami Fadali
Professor EBME
University of Nevada
Reno
1
Outline





Linear versus nonlinear.
Nonlinear behavior.
Controllability and observability.
Stability.
Passivity.
2
State Variables
 Minimal set of variables that
completely describe the system.
 State: set of numbers (initial
conditions) that allows us to solve for
the response for a given input.
 State variable: variables obtained by
letting the state evolve with time.
 Example: position, velocity.
3
Linear State-space Model
 State equations: Set of linear first-order
differential equations.
 Output equations: Set of algebraic
equation.
dx1
 x2
dt
x  f x, u, t
dx2
2


2
x
x

x
y  h x, u, t
1 2
2u
dt
y  x1  u




4
Linear State-space Model
 Linear equations.
 Can be solved
analytically.
x  Ax  Bu
y  Cx  Du
1   x1  0
 x1   0
 x    8  6  x   1u
 2   
 2 
y  1 1x
5
Linear Systems
 Additivity: add responses for added effects.
 Homogeneity: scale responses for scaled
effects.
 Zero-input Response: Due to initial
conditions.
 Zero-state Response: Due to the input.
 Total response = zero-input response +
zero-state response
At t 0 
x(t )  e
x(t0 )   e At   Bu( )d
t0
 

zero input
response
t
zero  state
response
6
Additivity & Homogeneity
Response to Initial Conditions
Step Response
3
0.5
0.45
2.5
0.4
2
0.35
0.3
Amplitude
Amplitude
1.5
1
0.25
0.2
0.5
0.15
0
0.1
-0.5
-1
0.05
0
0.5
1
1.5
2
Time (sec)
Zero-input response.
2.5
3
0
0
0.5
1
1.5
2
2.5
3
Time (sec)
Zero-state response.
7
Nonlinear Systems
x  f x, u, t 
y  hx, u, t 
x  f x, t   Gx, t u
y  hx, t   Dx, t u
 No additivity or homogeneity.
 Dependent responses due to initial
conditions and input.
 More complex behavior
8
Examples of Nonlinear Behavior
 Multiple equilibrium points.
 Limit cycles: fixed period without external
input.
 Bifurcation: drastic changes of behavior
with small changes in parameter values.
 Chaos: aperiodic deterministic behavior
which is very sensitive to its initial
conditions.
 Response to sinusoid: harmonics,
subharmonics or unrelated frequencies.
9
Multiple Equilibrium Points




Equilibrium: stay there if you start there.
Stability of equilibrium not system.
No change: time derivative is zero.
Solve for equilibrium points.
x  0  f x,u, t 
10
Examples
 Pendulum: Two
equilibrium points.
 Bistable Switch
 3 equilibrium points
(0, v0, v1)
dv
  g (v )
dt
g (v)  vv  v0 v  v1 
g(v)
v
11
Limit Cycles
 Unlike linear system
oscillations
2
1.8
1.6
1.4
x2
 Amplitude does not
depend on initial state.
 Stable or unstable limit
cycle.
Solution of limit cycle
2.2
1.2
1
dx1
 1  x1 x2
dt
dx2
  x1  1x2
dt
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
1.2
x1
1.4
1.6
1.8
2
12
2.2
Example: Fitzhugh-Nagumo Model
 Simplified version Param v0 v1
of H-H model.
0.2 1
Value
 Parameters
I
k
b
1
0.5
0
1.05
1.04
dv
  g (v )  w  I
dt
dw 1
 v  kw  b 
dt 
g (v)  vv  v0 v  v1 
1.03
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
13
Bifurcation
 Bifurcation point: Behavior d 2 x   x  x3  0
dt2
changes drastically as
3
parameter changes slightly.  xe  xe  0  xe  0, 
 Example: As parameter
xe
changes: periodic oscillations
Stable
 period doubling chaos.
 Example: Pitchfork
 Undamped Duffing
Equation
Stable
Unstable

Stable
14
Chaos
 Behavior is extremely sensitive to
initial conditions.
 Behavior is deterministic but looks
random.
 Example: cardiac arythmia (irregular
beating patterns)
15
Lorenz attractor
 Two unstable
equilibrium points.
 Model turbulent
convection in fluids
(weather patterns).
16
Response to Sinusoid
 Linear: scale amplitude and phase shift.
 Nonlinear:
 Harmonics: multiple of input frequency.
 Subharmonics: fraction of input frequency.
 Unrelated frequency.
 Examples
sin 2  x   1  cos2 x  2
sin  x  
1  cos2 x  2
17
Response to Noise
 Linear Systems
 Gaussian input gives Gaussian output.
 Completely characterized by mean and
covariance matrix (variance).
 Total response = zero-input response +
zero-state response
 Nonlinear systems
 Gaussian input gives non-Gaussian output.
 Need higher order statistics.
18
Example: Chi-Square Distribution
2



x

m

x
1
f X ( x)  2 1/ 2 exp 

2
2 x 
2 x 

x
t 2 2
1
2  

( x) 
e
fX(x)
dt
x
0.2
n
y   xi2 , xi ~ N 0,1
i 1
fY(y)
0.18
0.16
0.14

fY ( y)   2
0,
1
n/2
 ( n / 2)

(u )   t
0
y ( n / 2 ) 1e  y / 2 , y  0
y0
n=4 D.O.F.
0.12
0.1
0.08
0.06
e dt  u  1!, u integer
u 1  t
0.04
y
0.02
0
0
20
40
60
80
100
120
140
160
180
200
19
System Properties




Stability
Controllability
Observability
Passivity
20
Robustness
 Property holds over a specified subset of
parameter space.
 Sensitivity: local measure of robustness.
 Robustness w.r.t. noise and
disturbances.
21
Bode Sensitivity
per unit changein F
F F
S  Lim
 Lim
p 0 per unit changein p
p 0  p p
p
F

Lim
F p 0  p
F
p
p  F p  F 

 
F  p F   p  small  p
F F
 F   p S p , small  p
p
22
Example:Biochemical System
 Metabolite Xi is produced from substrate Xj
by an enzyme-catalyzed reaction (MM
Kinetics)
V
X
max
j
X i  v 
Km  X j
S
v
Xj

Km  X j  X j Km  X j
v X j


 Vmax

2
X j v
Vmax
K m  X j 
Km

Km  X j
23
Sensitivity Equation
 First-order estimates of the effect of
parameter variations (near q*)
x  f x, t , q 
S : x q
S  At , q S  Bt , q , S (t 0 )  0
f 
At , q   
x  q
f 
Bt , q   
q  q
24
Stability
 Local or global
 Lyapunov stability: continuity w.r.t. the
initial conditions.
 Asymptotic stability: Lyapunov
stability plus asymptotic convergence to
the equilibrium.
 Exponential stability: ||x|| trajectory
bounded above by an exponential decay.
25
Stability
2
1.8
Exponential
1.6
Unstable
1.4
Stability
1.2
1
0.8
0.6
0.4
Asympt.Stable
0.2
0
0
0.5
1
1.5
2
2.5
3
Stable i.s. L.
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Example: Model of Linear Pathway
 Specify kinetic orders, independent variables
 Determine equilibrium: (1/4, 1/16, 1/64)
 Solve differential equations (separation of
variables): asymptotically stable.
X4

X 1  0.5 X 4  X 10.5
X  0.5 X 0.5  4 X
2
1
X 3  4 X 2  2 X
X 4  0 .5
X1

2
0.5
3
X2
X3
Equilibrium: (0, 0, 0)
x1  X 1  1 4
x2  X 2  1 16
x3  X 3  1 64
0.5
x1  0.5x1  1 4
0.5
x2  0.5x1  1 4  4x2  1 16
0.5
x3  4x2  1 16  2x3  1 64
27
Stability of Motion
 Stability of equilibrium
of the error dynamics
2
1.5
1
x  f x, t 
x m  f m x m , t 
Error
0.5
0
e  x  xm
e  f x m  e, t   f m x m , t   f e e, t 
-0.5
0
0.5
1
1.5
2
2.5
3
28
Lyapunov Stability Theory
 Generalized energy function (positive
definite).
 Energy min at a stable equilibrium,
energy max at an unstable equilibrium.
 Trajectories converge to equilibrium if
energy is decreasing in its vicinity
(negative definite).
 Design: choose control to make
energy decreasing along trajectories.
29
Laypunov Stability Theorem
 Asymptotic stability
if
x  f (x)
V ( x)  0
T

V (x)  V x f  0
x(k  1)  f (x(k ))
V (x(k ))  0
V (x(k ))  V (x(k  1))  V (x(k ))  0
30
Lyapunov Approach
 Use quadratic
Lyapunov function.
 Local stability for v < 0.2
 Negative definite
f(v) derivative


1 2
v  40w2
2
dV
dv
dw
v w
dt
dt
dt
 v 4  1.2v 3  0.2v 2  vw  wv  0.5w2
V
v
dv
  g (v )  w
dt
dw 1
v  w 2

dt 40
g ( v )  v 3  1 .2 v 2  0 .2 v
0  v 4  1.2v 3  0.2v 2  0,0,1,0.2
31
Method to Obtain a Lyap Function
 Krasovskii’s method: Use Jacobian
(derivative) of RHS of state eqn.
 Stable if the derivative is negative near
the origin.
x  f (x)
V (x)  f T Pf  0
V (x)  f T AT Pf  f T PAf  f T Ff  0
F  AT P  PA  0
A  f x
32
Example: Metabolic Process
 Use Jacobian
(derivative) of
RHS of state eqn.
101 1 0
P   1 1 0
 0 0 1

1

 2 4 x1  1

1
A
 2 4 x1  1

0



0
0



4
0


4

4 
64x3  1 
0.5
x1  0.5x1  1 4
0.5
x2  0.5x1  1 4  4x2  1 16
0.5
x3  4x2  1 16  2x3  1 64
100



4 x1  1

F   4

0





8
4
0
16

4 
64x3  1 

4
0
33
Example: Fitzhugh-Nagumo Model
 System with zero bias has a stable
equilibrium (stable node) at (0,0).
 Small perturbation: return to equilibrium.
1.5
dv
  g (v )  w
dt
dw 1
v  w 2

dt 40
g (v)  vv  0.2 v  1
1
0.5
0
-1
-0.5
0
0.5
1
1.5
-0.5
0
0.5
1
1.5
-0.5
-1
-1.5
-1
34
Limitations
 Sufficient conditions for stability and
instability: if condition fails, no
conclusion.
 Necessary and sufficient for the linear
case only.
A P  PA  Q
T
A PA  P  Q
T
Q  0, P  0
35
Controllability & Observability
 Controllability: Can go wherever you want
no matter where you start.
 x0, xf ,  control u:[0,T]U, T < , s.t. x(T; x0) = xf.
 Indistinguishable:  u U
 x01, x02, y:[0,T]Y, T < 
 y(T, x01) = y(T, x02)
 Observability: Can determine the initial
state from the measurements (no two are
indistinguishable).
 x01, x02, y(T, x01) = y(T, x02)  x01= x02.
36
Graphical Interpretation
x2
Uncontrollable
Subspace
Controllable
Subspace
u
x2
Unobservable
Subspace
x1
Observable
Subspace
y
x1
37
Example
 Identical tanks with identical connections
to a water source.
 Not observable: Measuring the difference gives
zero regardless the two levels.
 Not controllable.: Filling the two tanks from one
source gives the same level.
38
Passivity




Supply rate: integrate to obtain energy.
Storage function: S
Dissipative system: storage < supply
Passive: dissipative with bilinear supply
rate.
S 0   0
S x(t )   0
S x(T )   S x(0)    uT ydt
T
0
T

S u y
39
Example of Passive System
 Spring-massdamper
 R-L-C circuit.
mv  bv   k x   f
vbv   0
k x   0
1 2
S  m v   k  x dx
2
dS dS dx dS dv


dt dx dt dv dt
 k  x v  m vv  v f  k  x   bv   k  x v
 vf  vbv   0
40
Zero Dynamics
 Internal dynamics of the system when the
output is kept identically zero by the input.
 Example: Metabolite Concentrations
 Select X4 such that X1 = 0 how do X2 & X3 behave?
X 1  0.8 X 21 X 31 X 40.5  3 X 10.5 X 20.1  2 X 10.75 X 30.2
X  3 X 0.1 X 0.1  2 X 0.5
2
1
2
2
X 3  2 X 10.75 X 30.2  5 X 30.5
X4  0
X 2  2 X 20.5
X  5 X 0.5
3
3
41
Stability of Passive Systems
 Zero-state detectable (observable)
System with zero input has stable
zero dynamics (resp. y=0  x=0)
 Theorem: Zero-state detectable and passive
a)  x=0 with u=0 is stable.
b)  x=0 with u= y=  h(x) is asymptotically stable.
42
Absolute Stability
(e)
 Stable for any
sector-bound
nonlinearity.
 G linear passive
e

e
u
G

 (.)
y

43
Example: Artificial Neural Networks
 Use passivity to show stability

V
1
b
e

Z-1
y

p

44
Passivity of Linear Systems (CT)
 A minimal state-space realization (A, B, C, D)
is passive if and only if there exist real
matrices P, L, and W such that
PP 0
T
AT P  PA  LT L
PB  C  L W
T
T
W W  DD
T
T
45
Passivity of Linear Systems (DT)
 A minimal state-space realization (A, B, C, D)
is passive if and only if there exist real
matrices P, L, and W such that
PP 0
T
AT PA  P   LT L
AT PB  C T  LT W
W W  D  D  B PB
T
T
T
46
Passivity of Periodic System
 (F, G, H, E) =DT minimal cyclic
reformulation of a periodic system.
 System is passive if and only if it
satisfies the following conditions
with
 a positive definite symmetric matrix P
 real matrices W and L.
47
Periodic KYP
Pi  AT (i) Pi 1 A(i)  Qi , i  0,, T 1, PT  P0
P0  T (T ,0) P0(T ,0)  Qs
T 1
Qs   T (i,0)Qi (i,0)
i 0
LTi Wi  C T (i)  AT (i) Pi 1 B(i), i  0,, T 1
WiT Wi  D(i)  DT (i)  BT (i) Pi 1 B(i), i  0,, T  1
48
Linearization
 Local behavior in the vicinity of an
equilibrium.
 Stability.
 Controllability.
 Observability.
 Passivity: KYP lemma.
49
Linearization
1st order approximation
f(x)
df
f ( x)  f ( x0 ) 
x  O(2 x)
dx x x0
x  x  x0
f  m x, for small x
x  f x 
 f x  
f x 
 i
 x
x 

x


x  x0
 x j x 
0 

80
f(x0)
60
40
20
0
0
2
4
6
x0
8
x
50
Linearization of Linear Pathway
Equilibrium: (1/4, 1/16, 1/64)
x  f x 
f x 
x 
x  Ax

x  x0

X 1  0.5 X 4  X 10.5
X  0.5 X 0.5  4 X
2
1
X 3  4 X 2  2 X 30.5
X 4  0 .5

2
 X 1   1 2 0
0   X 1 
   
 X 

X

1
2

4
0
 2 
 2 
 X 3   0
4  1 8  X 3 


Stable Equilibrium: (1/2, 4,  1/8) all in LHP
51
Stability
 Stability Condition:
Eigenvalues in LHP
dx1
 x2
dt
dx2
  x1  1 x2  1
dt
Equilib. x1  1 x2  0
x  f x 
f x 
x 
x  Ax

x  x0
1   z1 
 z1   0
 z    1  1  z 
 2 
 2 
2    1  0
52
Stability of Linear DT Systems
 Eigenvalues inside
the unit circle.
 Examples
x(k  1)   x(k )
 stable, a  1
x(k  1)  ax(k )  
unstable, a  1
Im[z]
Unit Circle
STABLE
Re[z]
UNSTABLE
0
0   x1 (k ) 
 x1 (k  1)  1 2
 x (k  1)  1 2  0.4
  x (k )
0
 2
 
 2 
 x3 (k  1)   0
4
 1 8  x3 (k ) 
53
Conditions for 2nd-order Case
 Second-order recursion
a2 x(k  2)  a1 x(k  1)  a0 x(k )  0,
a2  0
a2  a1  a0  0
a2  a0  0
a2  a1  a0  0
a0
1
stable
1
a1
1
1
54
Example: Dynamic Neural Network
 IIR Filter
 Nonlinear activation function (monotone increasing,
slope =g2 >0)
 Stable network for any stable matrix A.
 Problem: How to minimize error subject to the
stability constraint?
bi0
xi(k+1)
u
wi
+
z1
xi(k)
y
bi
+
f (.)
A
55
Constrained Optimization
 Minimize square error subject to stability
constraints.
 Consider 2nd-order (explicit constraints)
 Modify: stability margins (safety factor)
2
J  min  d i  ai 
a0 , a1
s.t.
i 1
2
1  a1  a0  0
1  a0  0
1  a1  a0  0
56
Global Linearization
 Find a transformation of the nonlinear
system to a decoupled linear system
(easy transformation is special
cases).
 Design linear control then transform
back.
 Use differential geometry to derive
the theory.
57
Example: Mechanical Systems
D(q)q
  C (q, q )q  g(q)  
q = vector of generalized coordinates.
D(q) = ss positive definite inertia matrix
C (q, q ) = ss matrix of velocity related terms
g(q) = s1 vector of gravitatioinal terms
 = vector of generalized forces
58
Global Linearization




Let the acceleration vector be the input.
Series of double integrators.
Choose acceleration for desired behavior.
Calculate torque from accelerations,
positions, and velocities.
(t )  u(t )
q
  D(q)u(t )  C (q, q )q  g(q)
59
Limitations
 Complex mathematical theory
(general case) but solution is hard to
obtain: must solve a nonlinear partial
differential equation analytically for
transformation.
 Results sensitive to modeling errors.
 Nonlinearity and coupling can be
exploited to provide desirable
behavior.
60
Discrete-time Periodic Systems
x(k  1)  Ad k x(k )  Bd k uk 
y(k )  C (k ) x(k )  D(k )u(k )
 All system matrices are governed by
M(k) = M(k+K) , k = 0, 1, 2, ...
 Model multi-rate sampled systems.
 Time-invariant reformulations: lift,
cyclic.
61
Fuzzy Models
 Include qualitative information.
 Fuzzy sets: graded membership.
Ai
Ai+1
…
…
x
ei1
ei
ei+1
62
Lyapunov Stability of TS Systems
 Linear matrix inequalities (LMI).
 Common Lyapunov function.
 Restrictive: system can be stable even if one or
more local model is unstable!
 Computational load: large number of LMIs.
A P  PA  0
T
A PA  P  0
T
Q  0
63
Hybrid Systems
 Switch between different models:
 Includes piecewise-linear.
 Overall behavior can be:
 Stable even if each subsystem is unstable.
 Unstable even if each subsystem is stable.
 Piecewise linear: Sufficient stability
condition using common Lyapunov
function.
64
Example:Gene Regulation
 Effector gene cycles between two
alternative environments:
 H = high demand environment (negative
control mode: repressor protein).
 L = low demand environment Positive
control mode: activator protein).
 TH (TL)Av. duration in phase H (L).
 Av. Cycle time = C = TH + TL
65
Mathematical Model
 AH x, x  H
x  
 AL x, x  L
x(t 0  TH )  exp(AH TH ) x(t 0 )
x(t 0  TH  TL )  exp(ALTL ) exp(AH TH ) x(t 0 )
x(t0  lC)   AHL  x(t0 )
l
 Response diverges if an eigenvalue of AHL is greater
than 1.
 Steady state: zero if all eigenvalues are inside the
unit circle.
 Nonzero for one or more eigenvalue on the unit circle.
66
Simulation Example
 0 1
1 0 
AH  
A

 L 1 0.1

1
1




1   x1 (k ) 
 x1 (k  1)   0
 x (k  1)   0.1 1.1  x (k )
 2 
 2
 
1  1, 2  0.1
1
v1    
1
x Lss  AL x Hss
1
v2    
0.1
1 0  1
1

    

1 0.1 1
1.1
67
Fault Detection
 Predict output of system using a
mathematical model.
 Compare predicted output to
measured output: primary residual.
 Filter the primary residual and use
the result to detect an error.
 Use state estimator (Kalman filter,
observer, Bayesian network, fuzzy
model, neural network)
68
Conclusion
 Mathematical models of physical systems
 Linear.
 Nonlinear.
 Piecewise-linear.
 Properties
 Stability.
 Controllability & observability.
 Passivity.
 Applications.
69
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